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の犬はとても大きい

someone's dog is very big

→ の
phrase Rare #40,064
someone's dog is very big の marks possession, so the phrase means “the dog belonging to someone is very big.” The possessor comes before の, and です may be added to make the sentence polite.
JA: 私の犬はとても大きいです。
EN: My dog is very big.

ケンさんの犬はとても大きい。

Ken's dog is very big.

君の犬はとても大きい。

Your dog is very big.

健の犬はとても大きい。

Ken's dog is very big.

うちの犬はとても大きいです。

My dog is very big.